Problem: All of the 4th grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$2.00$ each for students, and the group paid $$31.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$9.50$ each for students, and the group paid $$137.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2y = 31}$ ${26x+9.5y = 137.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-8y = -124}$ ${26x+9.5y = 137.5}$ Add the top and bottom equations together. $ 1.5y = 13.5 $ $ y = \dfrac{13.5}{1.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6.5x+2y = 31}$ to find $x$ ${6.5x + 2}{(9)}{= 31}$ $6.5x+18 = 31$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {26x+9.5y = 137.5}$ and get the same answer for $x$ ${26x + 9.5}{(9)}{= 137.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.